# Frequency Analysis

As shown in this tour, the notion of instantaneous
frequency should be handled with caution. This is why we recall the
primary definition of a frequency.

Note: this presentation is proper to the site and does not come
from the book.

## Period and Frequency

What is a frequency?

A frequency is the inverse of a period.

A one dimensional signal is T>0 periodic if it
is unchanged by a translation of T. Hence its support has an infinite
length. Nonetheless, the signal is entirely determined by its values
over an interval of length T.

In the
representation over an interval of length T, the *regularity * of
the signal implies the equality of the values of the functions and of
its derivatives at the left and right ends of the interval.

## A musical example

A purely synthetic music note can be represented
by a sinusoidal wave. An instrumental note that is held is a more
complex signal. A music or a speech recording is even more complex;
in particular, the frequencies may vary with time.

As a first approximation, a music piece may be
modelized as a sequence of signals defined over intervals whose
length is determined by the tempo. Every elementary signal can be
periodized.

## Fourier Analysis

Classically, the analysis of a signal as a Fourier
series or a Fourier integral provides a representation of its
frequency contents.

Fourier
series